Discovering Prince of Calculand

AP Calculus Group 2005-06 Story/Problem Project.

Sunday, November 20, 2005

Episode One: The Cerebus

After dusting himself, he collects himself and figures out he was in a dark, cold cave. He begans to walk about to where he though he had heard deep breathing. Harmless as the breathing sounds, he had not seen to who was breathing. A creature that he had seen and read only in books stood before him. Frightening as he imagined it could be. It was a cerebus. A three headed dog with razor teeth, a venomous snake for a tail, and a swarm of snakes growing on the dog’s back. It was the watchdog of the underworld.

He was dead, or was about to be. It was a fight for his consciousness. To regain it, he has to move Cerebus from a door it was blocking and somehow end up at the door without Cerebus knowing. The door would lead him out of this cave and his true consciousness back. He took a look around on the other end of the cave and planned. If he could aggravate the dog to at least move a foot from the door and use his didgeridoo (ancient flute) to put the dog to sleep, would it work?

If the dog’s three heads move simultaneously and the velocity is given by the function 0.5x^3 - 3x^2 + 0.25x + 5 m/min and the acceleration function is 1.5x^2 - 6x + 0.25. Find the following to help our young man escape the underworld in under 10 minutes or he could die:

1) When is the heads of the dog not moving?

2) At what time does our young man have to be most careful? (*Hint* Steep tangent slopes on parent function.)

Prologue

It was cold and dark, only the breathing of a young man that lies on the sand could be heard. He regains his consciousness and wonders where he was, how he got there and the strangest thought in his mind, who was he?